1. Hybrid Bonding Structure

Hybrid bonding integration can be broadly categorized into structures with 2 metal levels and 4 metal levels (Figs. 1(a)-1(c)). In the 2-metal-level bonding integration, the contact area is formed directly on the metal line. In contrast, the 4-metal-level hybrid bonding utilizes a 4-metal layer structure composed of Hybrid Bonding Metal (HBM) and Hybrid Bonding Via (HBV) at the Back End of Line (BEOL), directly connecting with other BEOL layers (Fig. 1(d)) ^[13]. This HBV structure enables the alignment of HBVs to desired positions while simplifying the process by connecting metal lines and HBVs in a single BEOL step. Moreover, by adopting the dual damascene process, it reduces misalignment issues and demonstrates robustness in fine-pitch hybrid integration ^[14-16]. Additionally, adjusting the structure and density of HBVs allows control over the current flowing through each metal, preventing void formation due to excessive current and enabling resistance tuning ^[17]. This paper proposes an effective equivalent resistance model for the structure in Fig. 1(c) to leverage these advantages of the HBV structure.

Fig. 2(a) presents a simple circuit model using dimensions and metal resistivity ^[12]. The dual damascene hybrid bonding structure is divided into three main parts in the equivalent circuit schematic: top metal, bottom metal, and HBV/HBM. Based on Eqs. (1) and (2) in Appendix, where the HBV is modeled as four vias connected in parallel, calculations using the actual structure's dimensions yield a total resistance of 170 m$\Omega$ for the analytical model in Fig. 2(a). However, FEM simulation of the same structure results in total resistance of 149 m$\Omega$, revealing a 12% error (see Appendix).

Notably, this calculation does not account for contact resistance, as no model for adding contact resistance was proposed. To analyze the causes of this significant error, a 4-metal bonding structure was created using Ansys Maxwell simulator ^[18,19]. The via size, metal thickness, and pitch used in this study were selected to reflect current trends in semiconductor packaging and practical process constraints. (See Table 1 in Appendix). By applying specific voltages to the top and bottom metals of the actual structure, current distribution and resistance components were examined. Fig. 2(b) shows the current density measured through Ansys FEM simulation for the 3D bonding structure. Using 11 vias, the simulation revealed noticeable deviations in current density across the vias. Specifically, the rightmost via on the top side of the hybrid bonding metal (HBM) exhibited the highest current density. Modeling all vias in parallel in the equivalent circuit leads to uniform current distribution, differing from FEM simulation results and causing errors. Thus, the parallel configuration of all vias in the existing equivalent circuit model introduces discrepancies, necessitating a refined circuit model calibrated with FEM simulation results.

Fig. 1. {Schematic diagrams of (a) 2-metal and (b-c) 4-metal hybrid bonding structures highlighting metal line contact regions and BEOL-level Hybrid Bonding Metal (HBM) and Hybrid Bonding Via (HBV) connections. (d) Illustration of the 4-metal hybrid bonding structure integrating HBM and HBV with other BEOL layers.}

Fig. 2. {(a) Equivalent circuit model using simplified dimensions and metal resistivity for the dual damascene hybrid bonding structure. (b) Current density distribution in the 3D hybrid bonding structure analyzed using Ansys FEM simulation, showing current variation across multiple vias.}

2. Proposed Circuit Model

Fig. 3(b) presents an improved model addressing the shortcomings of the analytical model in Fig. 3(a). The top metal's resistance is divided into multiple resistive components (R${}_{top.sub}$), allowing the equivalent model to represent varying potential within a single metal layer. This modification accurately reflects the FEM simulation result in Fig. 2(b), where current density is lowest in the leftmost via and highest in the via closest to the current source. Fig. 3(c) further divides the resistance components of the HBM bonding pad by width and length, classifying resistances based on current flow direction into core and substrate directions. Fig. 3(d) introduces a contact resistance component between HBMs.

According to previous studies, the contact resistivity at the Cu/Cu interface during bonding is approximately $10^{-8}$ to $10^{-9}$ $\Omega\cdot$cm$^2$ ^[12,20,21]. Assuming a void-free bonding interface, substituting this value into R${}_{\rm contact}$ should align the model's total resistance with electrical measurements of perfectly bonded structures. This adjustment effectively incorporates contact resistance into the model.

The proposed equivalent resistance model initially assumes vias distributed along the metal width in the X-axis direction, as shown in Fig. 4(a). However, it must also accommodate cases where the number of vias is smaller than the metal width due to process constraints, as in Fig. 4(b). To model resistance characteristics according to via count, FEM simulations and circuit model comparisons were conducted for HBV arrays ranging from $1\times 4$ to $5\times 4$ vias (Fig. 4(c)). Results show perfect agreement between the circuit model and FEM simulation for the $5\times4$ via configuration. However, in the $1\times 4$ array, FEM simulation yielded 149.913 m$\Omega$, while the proposed circuit model calculated 131.265 m$\Omega$, resulting in a 13% error.

This discrepancy stems from biased current density in the top metal due to the via distribution along the X-axis.

Fig. 4(b) indicates that the current density is highest near the center vias and decreases toward the outer metal edges. This suggests that directly using the top metal's cross-sectional area in the resistance formula is inappropriate for precise modeling. For accurate resistance calculation, a corrected cross-sectional area S', which adjusts the original area S by accounting for via distribution, should be used. To calculate this correction factor, FEM simulations were conducted on $1\times4$ via arrays while varying the top metal width from 5400 $\mu$m to 1200 $\mu$m (Fig. 4(d)). This process identified the correction factor that aligns the resistance values of the FEM simulation and the proposed model. Results indicate that the correction factor increases proportionally with the top metal width, improving the accuracy of the equivalent resistance model when factoring in the via count relative to the metal width.

Fig. 3. {(a) Previous analytical model of the hybrid bonding structure lacking accurate current distribution representation. (b) Improved equivalent circuit model with segmented top metal resistance (R${}_{\rm top.sub}$) to reflect potential differences within a metal layer. (c) Refined model separating HBM resistance components by width and length, classified into core and substrate directions. (d) Proposed enhanced model incorporating contact resistance at the Cu/Cu interface to improve alignment with experimental measurements.}

Fig. 4. {(a) Ideal via distribution along the X-axis, matching the metal width for accurate resistance modeling. (b) Non-uniform via distribution scenario, highlighting current density concentration at the metal center and reduction at the edges. (c) Comparison of FEM simulation and circuit model results for varying via configurations from $1\times4$ to $5\times4$ arrays. (d) FEM simulation-based correction factor determination for varying top metal widths in the $1\times4$ via configuration.}

1. ANSYS Simulation

The developed equivalent circuit model was thoroughly verified against FEM simulations under various structural and process parameters. Figs. 5(a)-5(c) compares the results of FEM simulations and circuit modeling for key parameters of vias in the hybrid bonding structure, including via count, height, and resistivity. In Fig. 5(a), the via count was increased from 15 ($5\times3$ array) to 25 ($5\times5$ array), and the total resistance values from both FEM simulations and the circuit model showed consistent agreement. Fig. 5(b) illustrates the comparison of resistance changes as via height varied from 550 nm to 1750 nm. Both models exhibited identical trends of increasing resistance with increasing via height, with minimal discrepancies in total resistance. Fig. 5(c) presents the comparison of resistance values as via resistivity changed. Both the FEM simulation and circuit model consistently followed the trend of increasing resistance with higher resistivity, confirming the high reliability of the proposed resistance model.

Additionally, Fig. 5(d) compares resistance values between FEM simulation and the circuit model by varying contact resistivity in the hybrid bonding structure. Contact resistivity is a critical parameter influenced by material and process conditions. In the FEM simulation, a 1 nm-thick resistive film was inserted at the bonding interface to emulate contact resistance. The resistance of this ultra-thin layer was controlled by adjusting its conductivity, and the contact resistivity and resistance were applied based on Eqs. (3) and (4). The results demonstrated excellent agreement between the two models, with discrepancies remaining below 1%. These findings validate that the proposed circuit model robustly and accurately predicts total resistance under variations in via height, via count, resistivity, and contact resistivity. This reliability suggests that the circuit model can efficiently optimize various parameters in hybrid bonding structure.

Fig. 5. {Comparison of total resistance between FEM simulation and circuit model (a) as the via count increases from a $5\times3$ to a $5\times5$ array, (b) with varying via heights from 550 nm to 1750 nm, (c) with varying via resistivity values, (d) by varying contact resistivity in the hybrid bonding structure.}

2. Fabrication and Electrical Measurement

To further verify the equivalent circuit model, experimental validation was conducted using fabricated devices. For experimental simplicity, a Dual Damascene Hybrid bonding structure was designed with horizontal alignment on the wafer, enabling adjustments in via count, via height, and top metal width for resistance comparison.

Fig. 6(a) shows an optical microscope image of the patterned structure after the lift-off process and just before measurements. The metal width was adjusted via the thickness of the deposited metal layers. In this experiment, a 50 nm Ti layer and a 200 nm Cu layer were deposited. To reduce contact resistance during probing, a 10 nm Ru layer was deposited subsequently to prevent Cu oxidation during later processes ^[3,22]. In the FEM simulation, a current of 100 mA was applied, and the voltage drop between the I${}_{\rm in}$ and I${}_{\rm out}$ (sink) terminals was measured.

The resistance was then calculated using Ohm's law. Similarly, in the actual electrical measurement, a constant DC current of 100 mA was applied using a two-point probe method, with one side set to ground. The voltage drop was measured across the probes, and the resistance was calculated based on Ohm's law, using a Semiconductor Parameter Analyzer (Keithley 4200A-SCS). However, the two-point probe method is affected by various parasitic resistances, including contact resistance, which can influence the measurement results. The parasitic resistance was measured to be approximately 0.2 $\Omega$, and this value was subtracted from the total measured resistance to obtain the corrected result.

As shown in Fig. 6(b), the experiment involved 16 structural configurations (Config.A, Config.B, Config.C) designed to analyze how via height, via count, and top metal thickness (d) impact the resistance of the dual damascene hybrid bonding structure. In Config.A, via height was varied incrementally from 69 $\mu$m to 552 $\mu$m across eight patterns to assess resistance trends. Fig. 7(a) confirms that measured values and circuit model results followed consistent trends with via height changes. Config.B involved increasing the via count from 2 to 5, with measurements again aligning with circuit model trends. Config.C further examined the impact of increasing the top metal thickness from 90 $\mu$m to 180 $\mu$m, producing results consistent with circuit model predictions. These experimental validations confirm the accuracy and robustness of the proposed equivalent circuit model across multiple parameter variations.

Fig. 6. {(a) Optical microscope image of the patterned structure after the lift-off process and prior to measurement. (b) Schematic of 16 structural configurations (Config.A, Config.B, Config.C) designed to evaluate the effects of via height, via count, and top metal thickness on resistance.}

Fig. 7. {Comparison of experimental measurements and circuit model results for resistance variation (a) with different via heights in Config.A, (b) with different via counts in Config.B and (c) top metal thickness in Config.C, demonstrating consistent trends.}